Continuing on, a point \(X\in \Gamma \)  will be given by coordinates \(X^{\alpha x} = \left(\rho (x),\Phi (x)\right)\) , where \(\alpha \)  takes on either of two values, \(\rho \)  or \(\Phi \) . (We use notation throughout that is consistent with [1]}.) Moreover, it is clear from the notation that \(\Gamma \)  is a doubly infinite-dimensional manifold: two-dimensions — one for \(\rho (x)\)  and another for \(\Phi (x)\)  — for each point \(x\in \mathbf {X}_{N}\) . As such, the dynamical system in question is also infinite-dimensional. This fact brings with it certain technical niceties (see e.g., [2]}), but by-in-large, we will assume that things are well-defined for the objects we are interested; when such subtleties are, in fact, relevant, we will be sure to mention them.
